Cepstrum
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A cepstrum (pronounced Template:IPA) is the result of taking the Fourier transform (FT) of the decibel spectrum as if it were a signal. Its name was derived by reversing the first four letters of "spectrum". There is a complex cepstrum and a real cepstrum.
The cepstrum was defined in a 1963 paper (Bogert et al.). It may be defined
- verbally: the cepstrum is the FT of the log (with unwrapped phase) of the FT
- mathematically: cepstrum of signal = FT(log(FT(the signal))+j2πm) (where m is the integer required to properly unwrap the angle or imaginary part of the complex log function)
- algorithmically: signal → FT → log → phase unwrapping → FT → cepstrum
The real cepstrum uses the logarithm function defined for real values, while the complex cepstrum uses the complex logarithm function defined for complex values also.
The complex cepstrum holds information about magnitude and phase of the initial spectrum, allowing the reconstruction of the signal. The real cepstrum only uses the information of the magnitude of the spectrum.
Many texts state that the process is FT → log → IFT, i.e., that the cepstrum is the "inverse Fourier transform of the log of the spectrum". This is not the definition given in the original paper, but it is widespread.
There are many ways to calculate the cepstrum; some of them need a phase-warping algorithm, others do not.
Applications
The cepstrum can be seen as information about rate of change in the different spectrum bands. It was originally invented for characterizing the seismic echoes resulting from earthquakes and bomb explosions. It has also been used to analyze radar signal returns.
It is now also used as an excellent feature vector for representing the human voice and musical signals. For these applications, the spectrum is usually first transformed using the Mel frequency bands. The result is called the Mel frequency cepstral coefficients, or MFCCs. It is used for voice identification, pitch detection and much more. Recently it has also been getting a lot of attention from music information retrieval researchers.
This is a result of the cepstrum separating the energy resulting from vocal cord vibration from the "distorted" signal formed by the rest of the vocal tract.
The cepstrum is also related to the homomorphic sound theory.
Cepstral concepts
The independent variable of a cepstral graph is called the quefrency. The quefrency is a measure of time, though not in the sense of a signal in the time domain. For example, if the sampling rate of an audio signal is 44100 Hz and there is a large peak in the cepstrum whose quefrency is 100 samples, the peak indicates the presence of a pitch that is 44100/100 = 441 Hz. This peak occurs in the cepstrum because the harmonics in the spectrum are periodic, and the period corresponds to the pitch.
Playing further on the anagram theme, a filter that operates on a cepstrum might be called a lifter.
References
- B. P. Bogert, M. J. R. Healy, and J. W. Tukey: "The quefrency alanysis of time series for echoes: cepstrum, pseudo-autocovariance, cross-cepstrum, and saphe cracking". Proceedings of the Symposium on Time Series Analysis (M. Rosenblatt, Ed) Chapter 15, 209-243. New York: Wiley, 1963.de:Cepstrum